Respuesta :
We have been an quadratic equation [tex]8k-k^{2} =42[/tex] and we are asked to solve our equation.
Upon subtracting 42 from both sides of our equation we will get,
[tex]8k-k^{2}-42 =0[/tex]
[tex]-k^{2}+8k-42 =0[/tex]
[tex]k^{2}-8k+42 =0[/tex]
Let us check whether our quadratic equation has any real roots using discriminant formula.
[tex]D=b^{2}-4ac[/tex]
Upon substituting our given values in discriminant formula we will get,
[tex]D=(-8)^{2} -4(1\cdot 42)[/tex]
[tex]D=64 -168[/tex]
[tex]D=-104[/tex]
We can see that D is less than zero, so our quadratic equation has no real roots.
Now we will use imaginary number i to find complex roots of our quadratic equation.
We will use [tex]-1 =i^{2}[/tex] to solve our equation.
[tex]k=\frac{-b \pm \sqrt{b^{2}-4ac} } {2a}[/tex]
[tex]k=\frac{--8 \pm \sqrt{(-8)^{2}-4(1*42)} } {2*1}[/tex]
[tex]k=\frac{8 \pm \sqrt{64-168 }} {2}[/tex]
[tex]k=\frac{8 \pm \sqrt{-1\cdot 104} } {2}[/tex]
[tex]k=\frac{8 \pm \sqrt{i^{2}\cdot 104} } {2}[/tex]
[tex]k=\frac{8\pm 2i \sqrt{26}}{2}[/tex]
[tex]k=4\pm i \sqrt{26}[/tex]
Therefore, our answer will be [tex]k=4\pm i \sqrt{26}[/tex].
